Question

If the radius of sphere is measured as 9m with an error of 0.03 m, then find the approximate error in calculating its surface area.

Answer 1

Let r be the radius of the sphere and $\Delta r$ be the error in measuring radius.
Then, r = 9 m and $\Delta r=0.03m$
Now, surface area of a sphere is given by $S=4\pi r^2\Rightarrow \frac{dS}{dr}=8\pi r$
Hence, $\Delta S=\left(\frac{dS}{dr}\right)\Delta r=\left(8\pi r\right)\Delta r=8\pi \times 9\times 0.03=2.16\pi m^2$
Hence, the approximate error in calculating the surface area is $2.16\pi m^2$